DUALITY QUANTUM COMPUTERS
Stan Gudder
Department of Mathematics
University of Denver
Denver, Colorado 80208
[email protected]
Abstract
We present a mathematical theory for a new type of quantum
computer called a duality quantum computer that has recently been
proposed. We discuss the nonunitarity of certain circuits of a duality
quantum computer and point out a paradoxical situation that occurs
when mixed states are considered. It is shown that a duality quantum
computer can measure itself without needing a separate measurement
apparatus to determine its final state.
1
Introduction
In a recent paper, Gui Lu Long proposed a new type of quantum computer
called a duality quantum computer [4]. According to Long, a duality quantum computer is much more powerful than an ordinary quantum computer.
In fact, a duality quantum computer can solve an unstructured database
search problem in logarithmic time and can solve NP-complete problems in
polynomial time. Moreover, Long has presented proof-in-principle designs
for two possible duality quantum computers. This indicates that if a general purpose quantum computer can be constructed, then a duality quantum
computer can probably also be constructed.
Simply stated, a quantum computer is a series of quantum gates, represented by unitary operators U1 , . . . , Un on a Hilbert space, that can be used
to perform a computation [1, 2, 3, 5]. An initial state ψ0 is input into the
1
quantum computer and then evolves into the output state ψ = Un · · · U1 ψ0 .
To gain information about the computation, we make a measurement on the
state ψ. If the measurement has m possible outcomes, then we obtain one of
these outcomes with a probability depending on the state ψ. This resulting
outcome gives information about ψ.
A duality quantum computer exploits the duality property that quantum systems can behave like both waves and particles. If a quantum system
evolves undisturbed then it acts like a wave and when it is observed or measured it acts like a particle. Now a quantum wave ψ can be decomposed into
parts using slits or beam splitters, for example. The wave parts or subwaves
can move along separate paths and then be combined at which point they
interfere. The subwaves are identical to ψ except they are at different locations along different paths. Because of these different locations, this does not
violate the no cloning theorem which says that an unknown quantum state
cannot be cloned exactly.
A duality quantum computer is a quantum computer that admits two new
operations, a divider operator and a combiner operator. The divider operator
decomposes the initial wave function into subwaves that are attenuated copies
of the initial wave evolving along different paths. Each of the paths can
contain quantum gates represented by unitary operators. After the subwaves
pass through the quantum gates they are collected together by the combiner
operator to form a final state. Finally, a measurement is performed on the
final state to gain information about the computation. These multiple paths
cause additional parallelism in a duality quantum computer and accounts for
their superiority over ordinary quantum computers.
This article provides mathematical details of some of the work in [4].
In particular, we discuss the nonunitarity of certain circuits in a duality
quantum computer. We also point out a paradoxical situation that occurs
when mixed states are considered for a duality quantum computer. Finally,
we show that a duality quantum computer can measure itself without needing
a separate measurement apparatus to determine its final state. In this paper
the states of a quantum system will refer only to the internal wave functions
and the position part of the wave functions will not be displayed. In this
way, the subwaves after a divider operation is applied will be copies of the
initial wave function except for an attenuation factor.
2
2
Divider and Combiner Operators
Let H be a complex Hilbert space and let p =(p1 , . . . , pn ) be a probability
distribution. That is, pi > 0, i = 1, . . . , n, and pi = 1. We use the notation
1/2
n
p = ( p2i ) and write H ⊕ for ⊕ni=1 Hi where Hi = H, i = 1, . . . , n. The
n
divider operator Dp : H → H ⊕ is defined by
Dp ψ =
1
⊕n (pi ψ)
p i=1
Thus, Dp maps ψ into attenuated copies of ψ. We think of each copy of H
n
in H ⊕ as a path
Lemma 2.1. The operator Dp is a linear isometry.
Proof. It is clear that Dp is linear and Dp is an isometry because
Dp ψ, Dp φ =
1
1 2
⊕(p
ψ),
⊕(p
φ)
=
pi ψ, φ = ψ, φ
i
i
p2
p2
We conclude from Lemma 2.1 that Dp is a unitary operator from H
onto its range R(Dp ). Of course, dim R(Dj ) = dim H. In particular, if
dim H = m < ∞ and {φ1 , . . . , φm } is an orthonormal basis for H then an
orthonormal basis for R(Dp ) is
1
1
(p1 φ1 , . . . , pn φ1 ), · · · ,
(p1 φm , . . . , pn φm )
p
p
We next define the operator C : H ⊕ → H by
n
C(ψ1 ⊕ · · · ⊕ ψn ) =
n
ψi
i=1
Again, C is a linear operator and we have that
C(ψ1 ⊕ · · · ⊕ ψn ), C(φ1 ⊕ · · · ⊕ φn ) =
=
ψi ,
φj
ψi , φj i,j
It follows that C is not an isometry. However, if we define Cp to be the
restriction of pC to R(Dp ) then we have the following result.
3
Lemma 2.2. The operator Cp is a linear isometry and Cp = Dp∗ .
Proof. To show that Cp is an isometry, we have that
Cp (p1 ψ ⊕ · · · ⊕ pn ψ), Cp (p1 φ ⊕ · · · ⊕ pn φ)
= p2 ψ, φ = p1 ψ ⊕ · · · ⊕ pn ψ, p1 φ ⊕ · · · ⊕ pn φ
To show that Cp = Dp∗ we have that
1
pi ψ = ψ
Cp Dp ψ = Cp
⊕ (pi ψ) =
p
Hence, Cp Dp = IH where IH is the identity operator on H.
We call Cp the combiner operator. Suppose we apply Dp and then a
unitary operator Ui on each of the paths, i = 1, . . . , n and finally apply Cp
to obtain
1
1
pi Ui ψ
⊕ (pi ψ) →
⊕ (pi Ui ψ) →
ψ → Dp ψ =
p
p
We call pi Ui a generalized quantum gate. Unlike an ordinary quantum
computer, a duality quantum computer admits generalized quantum gates.
We now give another way to describe generalized quantum gates. Define the
n
n
operator ⊕ni=1 Ui : H ⊕ → H ⊕ by
⊕Ui (φ1 ⊕ · · · ⊕ φn ) = U1 φ1 ⊕ · · · ⊕ Un φn
It is easy to check that ⊕Ui is unitary and
pi Ui = pC(⊕Ui )Dp
Denoting the set of generalized quantum gates on H by G(H), it is easy
to show that G(H) is a convex set. Of course, any unitary operator on H is
in G(H) and in particular IH ∈ G(H).We would now like to show that IH is
an extreme point of G(H). That is,
pi Ui = IH if and only if Ui = IH for
all i. It is not hard to show that this latter result holds when dim H < ∞
and the Ui mutually commute. Indeed, in this case the Ui are simultaneously
diagonalizable with matrix representations
Ui = diag (λi1 , . . . , λim )
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and since
pi Ui = IH we have that ni=1 pi λij = 1, j = 1, . . . , m. Hence,
pi Re(λij ) = 1 and it follows that Re(λij ) = 1 for all i, j. Since |λij | = 1
we have that λij = 1 for all i, j so that Ui = IH for all i. We now give a more
delicate argument to prove this result in general.
Theorem 2.3. The identity IH is an extreme point of G(H).
Proof.Suppose that
pi Ui = IH . Letting ψ ∈ H with ψ = 1 we have
that
pi Ui ψ = ψ. By Schwarz’s inequality we obtain
1 = ψ2 =
pi pj Ui ψ, Uj ψ ≤
pi pj |Ui ψ, Uj ψ|
≤
i,j
i,j
pi pj =
i
pi
i,j
pj = 1
j
Hence,
pi pj |Ui ψ, Uj ψ| = 1
i,j
Since |Ui ψ, Uj ψ| ≤ 1 we have that |Ui ψ, Uj ψ| = 1 for all i, j. But then
|Ui ψ, Uj ψ| = Ui ψ Uj ψ
so we have equality in Schwarz’s inequality for all i, j. Hence, Ui ψ = αUj ψ
with |α| = 1. Thus, there exist αi ∈ C with |αi | = 1 such that Ui ψ = αi U1 ψ
for all i. We conclude that
ψ=
pi Ui ψ =
pi αi U1 ψ
Therefore, for any unit vector ψ there exists an αψ ∈ C with |αψ | = 1
such that U1 ψ = αψ ψ; that is, every nonzero vector is an eigenvector of U1 .
The only unitary operators with this property are αIH , |α| = 1. Hence,
U1 = α1 IH with |α
1, i = 1, . . . , n.
1 | = 1 and similarly
Ui = αi I with |αi | =
Therefore, IH =
pi αi IH so that
pi αi = 1. But then
pi Re(αi ) = 1
so that Re(αi ) = 1, i, . . . , n. Hence, αi = 1 and we conclude that Ui = IH ,
i = 1, . . . , n.
Corollary 2.4. The extreme points of G(H) are precisely the unitary operators H.
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Proof. Suppose A ∈ G(H) is an extreme point. Then A =
pi Ui which
implies that Ui = A for all i so A is unitary. Conversely, supposeU is unitary.
To show that U is an extreme point of G(H) assume that U = pi Ui . Then
∗ IH = U U ∗ =
p i Ui
pi pj Ui Uj∗
pj Uj =
i,j
Applying Theorem 2.3 we conclude that Ui Uj∗ = IH for all i, j. In particular,
Ui = U1 for all i. Hence,
U=
pi U1 = U1 = Ui
for all i.
We conclude from Corollary 2.4 that except for a degenerate probability
distribution, no generalized quantum gate is unitary; that is, no generalized
quantum gate is a quantum gate. Stated in another way, except for the
case of a single path no duality quantum computer can be described by an
ordinary quantum computer. In a sense, a duality quantum computer is a
mixture of ordinary quantum computers.
An example of a generalized quantum gate occurs in the following vector
selection algorithm. Let ψ1 , . . . , ψN be an orthonormal basis for H and
suppose we want to select a marked but unknown vector ψk from among
them. A quantum computer possess an oracle (black box) that recognizes ψk
i,k ψ .
and the oracle is given by the unitary operator U where U ψi = −(−1)δ
i
Let p be the probability distribution p = (1/2, 1/2) and let ψ = (N )−1/2 ψi
be the input state for a duality quantum computer. Form the generalized
quantum gate given by
1
1
p C(I ⊕ U )Dp = IH + U
2
2
Since 12 (IH + U )ψi = δi,k ψi we have that 12 (IH + U ) = Pk where Pk is the
projection onto the one-dimensional subspace generated by ψk . Moreover,
1
(I + U )ψ = (N )−1/2 ψk so the duality quantum computer selects the marked
2
vector ψk using a single query to the oracle. This is the mechanism behind
Long’s logarithmic time database search algorithm [4].
We have seen that G(H) is a convex set whose extreme points are the
unitary operators on H. Let B(H) be the set of bounded linear operators on
H and let R+ G(H) be the positive cone generated by G(H). That is,
R+ G(H) = {αA : A ∈ G(H), α ≥ 0}
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The next result shows that if dim H < ∞ then a duality quantum computer
can simulate any operator on H.
Theorem 2.5. If dim H < ∞, then B(H) = R+ G(H).
Proof.
For A ∈ B(H) we must show that there exists an α ≥ 0 such that A =
α pi Ui where Ui are unitary and (p1 , . . . , pn ) is a probability distribution.
If A = 0, then letting α = 0 we are finished, so suppose A = 0. Let P ∈ B(H)
be a projection operator. Let ψ1 , . . . , ψM be an orthonormal basis for R(P )
and extend to an orthonormal basis ψ1 , . . . , ψN for H. Define the unitary
operator U on H by
ψi if i = 1, . . . , M
U ψi =
−ψi if i = M + 1, . . . , N
Then P = 12 IH + 12 U ∈ G(H). Now letA ∈ B(H) be self-adjoint with
distinct eigenvalues λi = 0 and let a =
|λi |. Let Pi be the projection
onto the eigenspace for λi . Then there exist unitary operators Ui such that
Pi = 12 IH + 12 Ui . Hence,
λi
|λi |
Pi = a
(sgnλi )Pi
A=
λ i Pi = a
a
a
|λi |
1
1
=a
sgnλi
IH + Ui
a
2
2
|λi |
|λi |
=a
(sgnλi )IH +
(sgnλi )Ui
2a
2a
Now (sgnλi )IH and (sgnλi )Ui are unitary and a−1 √ |λi |= 1. Hence, A has
the required form and of course, so does iA i = −1 . Now suppose A
is an arbitrary operator on H. Then A = B + iC where B and Care selfadjoint. Now from before we have that B = b pi Ui and C = c qi Vi in
the required form. Thus,
1
1
A = B + iC = (b + c)
B+
iC
b+c
b+c
b c = (b + c)
pi Ui +
qi Vi
b+c
b+c
Since
c b pi +
qi = 1
b+c
b+c
we are finished.
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Corollary
2.6. If dim H < ∞, then
A ∈ B(H) is normal if and only if
A = α pi Ui where α ≥ 0, pi > 0,
pi = 1 and Ui are unitary operators
that mutually commute.
3
Mixed States
Suppose we have
a duality quantum computer represented by the generalized
quantum gate
pi Ui . If the input state is represented by a unit vector ψ,
then presumably the output state is represented by the unit vector
pi Ui ψ pi Ui ψ Notice
that
it
was
necessary
to
renormalize
the
vector
pi Ui ψ because
pi Ui is not unitary in general. Instead of a pure state, suppose we input a mixed state represented by a density operator ρ. In accordance with
the formalism of quantum mechanics, the divider operator Dp will transform
ρ to the state Dp ρDp∗ = Dp ρCp . Since
Dp ρCp [⊕(pi φ)] = p Dp ρ
pi φ = p Dp ρφ
= ⊕(pi ρφ) = (⊕pi ρ)φ
we conclude that Dp ρDp∗ = ⊕pi ρ. If we now apply the quantum gates ⊕Ui
along thevarious paths we obtain ⊕pi Ui ρUi∗ . Finally, applying the operator
C gives
pi Ui ρUi∗ . The map E(ρ) =
pi Ui ρUi∗ is called a quantum operation in the
literature [1, 2, 3, 5]. One advantage of this approach is that E(ρ) is again a
state so we do not have to renormalize. Indeed, clearly E(ρ) is positive and
we have that
tr [E(ρ)] = tr
pi Ui ρUi∗ =
pi tr (Ui ρUi∗ )
=
pi tr(ρ) = 1
Unfortunately, even when ρ is pure, E(ρ) is pure only under very special
circumstances as the next result shows. The one-dimensional projection onto
the subspace spanned by a unit vector ψ is denoted by Pψ . Of course, Pψ is
a density operator representing the pure state ψ.
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Theorem 3.1. The state E(ρ) is pure if and only if ρ = Pψ is pure and there
exists a unit vector φ and αi ∈ C, |αi | = 1 such that Ui ψ = αi φ for every i.
Proof. If the condition holds then
pi Ui Pψ Ui∗ =
pi PUi ψ
E(ρ) =
pi Ui ρUi∗ =
=
p i Pφ = P φ
Hence, E(ρ) is pure. Conversely, suppose E(ρ) is pure so that E(ρ) = Pφ .
Then
pi φ, Ui ρUi∗ φ = φ, E(ρ)φ
pi Ui∗ φ, ρUi∗ φ =
= φ, Pφ φ = 1
It follows that Ui∗ φ, ρUi∗ φ = 1 for every i. Hence, ρ = Pψ for some unit
vector ψ and ρ is pure. Moreover,
φ, PUi ψ φ = φ, Ui Pψ Ui∗ φ = 1
for every i. We conclude that PUi ψ = Pφ so that Ui ψ = αi φ with |αi | = 1 for
every i.
It follows from Theorem 3.1 that this approach for mixed states
is not a
generalization of the previous approach where we had ψ → ( pi Ui ) ψ. In
fact, we can show this directly as follows. If ρ = Pψ is a pure state then
E(ρ) = E(Pψ ) =
pi Ui Pψ Ui∗ =
pi PUi ψ
which is a lot
different
than
(
pi Ui ) ψ. The two approaches are equivalent if
and only if
pi PUi ψ is a one-dimensional projection for every unit vector ψ.
But this holds if and only if all the Ui coincide up to multiplicative constants
of modulus 1. But then
pi PU1 ψ = PU1 ψ
E(ρ) =
Pi U1 Pψ U1∗ =
which is an ordinary quantum computer.
This gives a definite puzzle of how to treat mixed states in a duality quantum computer. If we are to truly understand duality quantum computers,
this paradox must be resolved. Notice that for ordinary quantum computers there is no such paradox. In this case there is only one path and pure
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states are transformed by ψ → U ψ while mixed states are transformed by
ρ → U ρU ∗ . Representing a pure state ψ by Pψ we have
Pψ → U Pψ U ∗ = PU ψ
which is equivalent to U ψ. Thus, the two approaches are equivalent for an
ordinary quantum computer.
Moreover, in this second approach we seem to lose the greater power contained in the first approach. For example, in the vector selection algorithm
we have that U1 = IH and U2 ψi = −(−1)δi,k ψi as before. Again, if the initial
state is the uniform superposition ψ then
E(Pψ ) =
1
1
1
1
PU1 ψ + PU2 ψ = Pψ + PU2 ψ
2
2
2
2
If we now measure the basis ψ1 , . . . , ψN then we have N possible outcomes
each with probability
1
1
1
1
1
|ψ, ψi |2 + |U2 ψ, ψi |2 =
+
=
2
2
2N
2N
N
This gives no information at all. In general, the state E(ρ) =
pi Ui ρUi∗ is
a mixture of states on each path so E(ρ) is a mixture of states each of which
evolves according to an ordinary quantum computer. Thus, E(ρ) cannot
give more information about the result of a computation than an ordinary
quantum computer.
Long mentions that the decomposition into subwaves can be iterated so
that any subwave can be further decomposed into sub-subwaves [4]. In this
way, more complex duality quantum computers can be constructed. However, as we now show, these iterations do not give anything new that cannot
already be accomplished with a single decomposition into subwaves. For
example, after producing ψ → p−1 ⊕ pi Ui ψ suppose we apply a divider
operator to the first path to obtain
m
n
1 1
qj Vj U1 ψ ⊕
pi Ui ψ
p1
p| q j=1
p i=2
E(Pψ )ψk , ψi =
After applying the combiner operator we obtain
p1
m
qj Vj U1 ψ +
j=1
n
i=2
10
pi Ui ψ
But this can be obtained using the single generalized quantum gate
m
p1 qj Vj U1 +
j=1
4
n
pi Ui
i=2
Projective Measurements
This section shows that projective measurements can be directly incorporated
into a duality quantum computer. To be precise, we show that a projective
quantum measurement can be performed using a generalized
quantum gate.
A general quantum operation has
the form E(ρ) = Ai ρA∗i where Ai are
arbitrary operators on Hsatisfying
A∗i Ai = IH . If the Ai are projection
operators Pi satisfying
Pi = IH , then E is called a projective measurement. Notice that a generalized quantum gate also gives a quantum
operation because we can write
√
√
E(ρ) =
pi Ui ρUi∗ =
pi Ui ρ pi Ui∗
where
√
√
pi Ui∗ Ui =
pi IH = IH
( pi Ui )∗ ( pi Ui ) =
Theorem 4.1. Let dim H < ∞ and let E(ρ) =
Pi ρPi be a projective
measurement.
Then
there
exists
a
generalized
quantum
gate
pi Ui such
∗
that E(ρ) = pi Ui ρUi .
Proof. We shall employ the unitary
theorem [5]
freedom
which∗ states that
∗
two quantum operations E(ρ) =
Ai ρAi and F(ρ) =
B
i ρBi coincide if
and only if there √
exists a unitary matrix [ujk ] such that Ej = k ukj Fk for all
i, j. Letting i = −1 , the discrete
Fourier transform is given by the unitary
−1/2 2πijk/n
n × n matrix n
e
. Define the unitary operators Uj , j = 1, . . . , n,
by
Uj =
n
e2πijk/n Pk
k=1
We can then write
1
1
√ Uj =
√ e2πijk/n Pk
n
n
k=1
n
11
Applying the unitary freedom theorem, we conclude that
∗ 1
1
1
√ Ui ρ √ Ui =
Ui ρUi∗
E(ρ) =
Pi ρPi =
n
n
n
References
[1] M. Brooks, Quantum Computing and Communications, Springer-Verlag,
London, 1999.
[2] J. Gruska, Quantum Computing, McGraw-Hill, London, 1999.
[3] M. Hirvensalo, Quantum Computing, Springer-Verlag, Berlin, 2001.
[4] G. L. Long, The general quantum interference principle and the duality
computer, arxiv: quant-ph/0512120, 2005.
[5] M. Nielsen and J. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.
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